THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 77 



the same in form a$ those from which DARWIN obtains the conditions of equilibrium. 



a 

 The equation ^- (<5E) = 0, written out in full, becomes 



......... (168) 



If second-order terms are neglected equation (168) reduces to 



a = 0, ........... (169) 



and if a is taken equal to zero, equation (168) reduces to 



.......... (170) 



DARWIN'S method is in effect to replace equation (168) by equations (169) and (170). 

 This, in my opinion, introduces one limitation too many on the values of the variables, 

 and so reduces the doubly-infinite series of figures of the second order to two singly- 

 infinite series. Equation (169) must undoubtedly be true if second-order terms are 

 neglected, but we may take 



....... (L7l) 



where X is a quantity of the second order entirely at our disposal, and still satisfy all 

 the conditions necessary for equilibrium. I think it will be found that DARWIN'S 

 procedure in effect introduces the spurious condition X = 0. DARWIN'S equations are 

 of course sufficient to ensure equilibrium ; what is maintained is that they are not 

 necessary and so do not disclose all possible figures of equilibrium. 



In this way I believe it will be found that DARWIN has limited himself to one linear 

 series of figures (X = 0) instead of the doubly-infinite series represented inside the 

 rectangle ABCD in fig. 1. 'If this series should happen to run on continuously at 

 the edge of the rectangle with the true series ROE/ in fig. 1 , then of course DARWIN'S 

 investigation would stand. But no reason suggests itself, and certainly DARWIN 

 (not foreseeing the complication of the doubly-infinite series) gives no reason, why this 

 should be the case. For some value of X the two series will run on continuously, but 

 there is no assignable reason why this value should be X = 0. 



What, then, will happen if we try to carry DARWIN'S approximation on to third- 

 order terms fjy his method 1 I think it will be found that his series comes to a dead 

 end before the third-order terms are arrived at, precisely as if it ran on to the 

 boundary of the rectangle ABCD in fig. 1, and could get no further. If the displace- 

 ment goes as far as third-order terms, SE must go as far as sixth-order terms, and 

 will contain terms of orders 2, 4, and 6, say 



